Partial orders on representations of algebras

Let k be a commutative artin ring and let Λ be an artin k-algebra. For each natural number d let repdΛ be the set of isomorphism classes of Λ-modules with k-length equal to d. For each natural number n, an (n×n)-matrix with entries in Λ, can be considered as a k-endomorphism of Mn, where Mn denotes the direct sum of n copies of the Λ-module M. The quasi-order ⩽n on repdΛ is defined by M⩽nN if for every (n×n)-matrix φ, with entries in Λ, we have that ℓk(Mn/φMn)⩽ℓk(Nn/φNn).We show that the quasi-order ⩽n is a partial order on repdΛ for n⩾d3.